Fixed points of abelian actions on S2

Abstract

AbstractWe prove that if ${\mathcal F}$ is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of $\mathbb {R}^2$ which leaves invariant a compact set then there is a common fixed point for all elements of ${\mathcal F}$. We also show that if ${\mathcal F}$ is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of ${\mathcal F}$ with index at most two.